Yes, the National Parks & Memorials did remain open Saturday

from NEWS posted at Obama Food-o-rama (Google Blog)

After a brief speech late on Friday night announcing that the government would remain "open for business" thanks to a budget deal that averted a federal shutdown, President Obama proved his point today with an "unscheduled" drop-in to the Lincoln Memorial, where he shook hands with thrilled tourists (above).

“Hi everybody. I just wanted to say, real quick, that because Congress was able to settle its differences, that's why this place is open today and everybody’s able to enjoy their visit," President Obama said to the crowd gathered at the top of the monument, in front of the seated stone statue of President Abraham Lincoln.
The President arrived at the monument to one of his own personal heroes shortly after 2:35 PM, casually attired in gray chinos and a black zip-up jacket. He bounded up the steps, surrounded by his Secret Service detail. Saturday was a busy day on the Washington, DC tourist calendar: The annual Cherry Blossom Festival Parade was held, as was the annual Spring Garden Tour at the White House.
The President could just as easily have mingled with the hundreds of tourists who were at that very moment romping around on the South Lawn, but his visit to the Lincoln Memorial twinned nicely with his speech last night; he noted then that the Washington Monument, and National Parks across the US, would remain open, thanks to the budget deal.